The 95% Rule says that if you go out two standard deviation units either side of the mean, you will capture 95% of cases. To get the scores that fall two standard deviations from the mean, simply multiply the standard deviation by two and subtract that from the mean to get the lower bound, and add it to the mean to get the upper bound.
For example, let us return to our example test where the mean was equal to 100, and the standard deviation was equal to 10. To find the score that defines the second standard deviation below the mean, we multiply 10 x 2 to get 20. We then subtract 20 from 100 (100 – 20 = 80) to get 80. We then add 20 to 100 (100 + 20 = 120) to get 120. This means that 95% of those taking the test had scores falling between 80 and 120.
95% Rule About 95% of cases lie within two standard deviation unit of the mean in a normal distribution.
The 99.7% Rule says that 99.7% (nearly all) of cases fall within three standard deviation units either side of the mean in a normal distribution. To get the scores that fall three standard deviation units from the mean, simply multiply the standard deviation by three and subtract that from the mean to get the lower bound, and add that to the mean to get the upper bound.
99.7% Rule About 99.7% of cases lie within three standard deviation unit of the mean in a normal distribution.
The 95% Rule is an approximation. To be more precise, you need to go out 1.96 standard deviation units on either side of the mean to capture 95% of cases. This seemingly strange number is dictated by the mathematics of the normal curve. Unfortunately, you can have nice round percentages, or you can have nice round standard deviation units, but you cannot have both. The properties of the normal curve just do not allow for it. The approximation is best used for seeing a distribution of scores in your mind’s eye. When further statistical analysis is involved, and your results will appear in reports, it is better to use the more precise multiplier.
For example, let us consider an IQ test that has a mean of 100 and a standard deviation of 15. To obtain the score that marks one standard deviation to the left of the mean, we multiply the standard deviation by the constant 1.96 (15 x 1.96) to get 29.4. We then subtract 29.4 from the mean of 100 to get 70.6. To find the upper bound, we add 29.4 to 100 to get 129.4. Thus, we can say that 95% of people will have an IQ score between 70.6 and 129.4.
If we go out 3 standard deviation units on either side of the mean, we will capture 99.7% of all cases. Given a choice between a round percentage and a round standard deviation unit, most researchers will choose to have a round percentage of 99.00%. This means that we have to use a multiplier that shaves off that 0.7% leftover. The 99% Rule says that if you go out 2.58 standard deviation units on either side of the mean, you will capture precisely 99% of cases.
To review which multiplier to use when working with a particular percentage, consult the table below. These constants are used so frequently in statistics, it is best to commit them to memory.
Percentage of Cases Multipliers
|To Obtain This Percentage of Cases…||Multiply The SD By This…|
Normal Curve, Gaussian Distribution, Bell Curve, Symmetrical
Last Modified: 06/03/2021